User p c - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T03:06:59Z http://mathoverflow.net/feeds/user/11454 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/48771/proofs-that-require-fundamentally-new-ways-of-thinking/48874#48874 Answer by P C for Proofs that require fundamentally new ways of thinking P C 2010-12-10T04:05:11Z 2010-12-10T04:05:11Z <p>Some more proofs that startled me (in a random order):</p> <p>Liouville theorem to prove that Weierstrass P-function satifies the differential equation you know.</p> <p>Complex methods to establish the addition law on an elliptic curve.</p> <p>Cauchy's formula (for P'/P) to prove that C is algebraically closed.</p> <p>Pigeon hole principle to prove existence of solutions to Fermat-Pell's equation</p> <p>Kronecker's solution to the same equation, using L-functions.</p> <p>Minkowski's lemma (a convex compact, symmetric, of volume 2^n contains a non trivial integer point) and its use to prove Dirichlet's theorem on the structure of units in number fields.</p> <p>Fourier transform to prove (versions of) the central limit theorem.</p> <p>Multiplicativity of Ramanujan's tau function via Hecke operators.</p> <p>Poisson formula and its use (for example, for the functional equation of Riemann's zeta function, or for computing the volume of SL_n(R)/SL_n(Z), or values of zeta at even positive integers).</p>